Application of the Swarm Intelligence Algorithm for Reconstructing the Cooling Conditions of Steel Ingot Continuous Casting

Abstract

This paper presents a proposal to apply one of the swarm intelligence algorithms, the artificial bee colony (ABC) algorithm, to solve the inverse problem of steel ingot continuous casting. The discussed task consists of retrieving the cooling conditions of the process on the basis of temperature measurements and by taking into account the macrosegregation phenomenon. The examined process was modeled by using the mathematical model of solidification within the temperature interval. The solution method was based on the implicit scheme of the finite difference method supplemented by the procedure of correcting the field of temperature in the vicinity of liquidus and solidus curves, which was then used for solving the appropriate direct problem. The computational example, illustrating the stability and accuracy of the proposed method, is also presented in the paper.

1. Introduction

The possibility of controlling thermal processes is an essential requirement in many industrial applications. One of the most important tools, used in modern preparations for industrial production, is delivered by computer programs simulating thermal processes. In order to create these kind of computer programs, it is necessary to provide efficient modeling of these processes as well as solving the direct and inverse solidification problems. The inverse problems are used when some input data of the process are unknown or not completely defined and one needs to reconstruct the missing input information on the basis of products resulting from the investigated process (see for example Beck et al. [1] and Alifanov [2]). By solving the inverse solidification problems one can choose the boundary conditions or the parameters of the solidifying material in a specific way in order to ensure the assumed run of the solidification process, thanks to which one can control the quality and properties of the resultant ingot.

Solving the inverse problem is obviously much more difficult than solving the direct problem in which all the initial data and conditions are known. However, in literature one can find several proposed approaches for solving the inverse problems concerning the heat transfer and solidification (see for example Zabaras et al. [3], Momose et al. [4], Nejad et al. [5] and Okamoto et al. [6]). In particular, in the work by Du et al. [7] an inverse algorithm is applied to calculate the heat flux between the ingot and the crystallizer on the basis of temperatures measured by using the thermocouples buried in the mould. The data obtained are applied to a full-scale finite element model of a slab and its mould and this model is used to analyze the non-uniform thermal and mechanical states of the slab. Next, the paper by Luo et al. [8] focused on estimating the heat transfer coefficients with the aid of the measured surface temperatures containing large disturbances. For the solution of this problem, the authors propose an integrated approach based on the Gaussian kernel function and the particle swarm optimization algorithm. The direct problem is solved with the aid of the finite difference method. The model discussed is two-dimensional and is transformed to the one-dimensional unsteady heat problem including the substitute thermal capacity. Moreover, the model assumes the linear solid phase fraction. A similar problem is investigated by Wang et al. [9], but a different optimization method is applied to find the solution. That method is the weighted least squares Levenberg–Marquardt method.

The secondary cooling heat transfer in the inverse heat transfer calculation model with temperature measurements is reconstructed by Wang et al. [10]. The proposed method transforms the inverse problem, consisting of parameter identification, into the optimization problem solved by using the evolutionary algorithm. In this paper the two-dimensional unsteady heat conduction problem is discussed. The linear solid state fraction is assumed and the effective value of thermal conductivity is used. Wang et al. [11] present the two-level parallel solution method implemented on a graphics processing unit (GPU) for identification of the heat transfer coefficients in continuous casting. The two-level parallel solution method consists of the parallel-based heat transfer model and the stream parallel particle swarm optimization. In the work by Wang et al. [12] the particle swarm optimization algorithm is applied for determining the heat transfer coefficient in the secondary cooling zone. The two-dimensional unsteady heat conduction problem with the substitute thermal capacity is examined in this paper. This is done by assuming the linear solid state fraction and the effective value of thermal conductivity in the liquid phase.

Yu et al. [13] reconstructed the heat transfer coefficient in the secondary and air cooling zone of the continuous casting. This paper combines the weighted least square method and the improved differential evolution algorithm to approximate the sought parameter. In this paper the two-dimensional unsteady heat conduction problem with the substitute thermal capacity is investigated and the implicit scheme of the finite difference method is applied for calculations. The study presented by Udayraj et al. [14] is devoted to estimating the boundary heat flux across the hot surface of the billet mould in contact with the molten steel during the continuous casting process. The model is based on the two-dimensional steady-state inverse heat transfer technique that is solved by using the conjugate gradient method. The examination of the heat transfer coefficients along the secondary cooling zone in the continuous casting of steel is also presented by Santos et al. [15], and in papers by Constales et al. [16] and Slota [17] the selection of the cooling strategy is executed for the primary and secondary cooling zone.

Chakraborti et al. [18,19] used the genetic algorithm for modeling the mould, spray and radiation cooling regions of the continuous ingot. The genetic algorithm is also applied for modeling the continuous casting regions by Cheung et al. [20], Santos et al. [21,22] and Slota [17]. Santos et al. [23] used a neural network based algorithm for this purpose in connection with the knowledge of operational parameters based on the boundary conditions and the metallurgical constraints. Next, in the paper by Vasko et al. [24] the water spray setting in continuous casting is optimized. The task is formulated as a non-linear programming problem, the objective of which is to minimize the sum of the cost penalties under assumption of the appropriate constraints.

Vasko et al. [24] also investigated the inverse solidification problems of pure metals (e.g., Hetmaniok et al. [25]) and of alloys (e.g., Hetmaniok [26]), as well as of continuous casting (e.g., Slota [17] and Hetmaniok et al. [27]) and in order to improve the procedure of the solution and speed up the process of calculations they used optimization algorithms used in artificial intelligence. In the current paper, these modern algorithms are applied for solving the inverse problem of continuous casting of the steel ingot. Examples of applying the artificial intelligence algorithms for controlling the continuous casting of steel can also be found in this study.

The goal of this paper is to present an algorithm dedicated to reconstructing the cooling conditions of the continuous steel ingot. The process examined is described by using the mathematical model of solidification in the temperature interval with only the distribution of temperature taken into account (see Mochnacki et al. [28,29,30] and Majchrzak et al. [31]). This model is based on the heat conduction equation with the source element enclosed that includes the latent heat of fusion and the volumetric contribution of the solid phase. For the given form of the function defining this contribution, the equation used transforms into the heat conduction equation with the substitute thermal capacity (see Majchrzak et al. [32]). The model investigated the effect of convective heat flow in the liquid region is taken into account by using the effective thermal conductivity for molten steel (see Lally et al. [33], Choudhary et al. [34] and Oksman et al. [35]). However, the other phenomenon influencing the solidification process, i.e., macrosegregation, is included by using Scheil’s equation (see Mochnacki et al. [30], Majchrzak et al. [31] and Santos et al. [22]).

The macrosegregation phenomenon consists of the differences of chemical composition occurring in the cast, which can reveal, for example, in such a way that the variation of concentration of the alloy component is accompanied by the variation of liquidus and solidus temperatures. In the result of the applied approach, the authors of the current paper intended to retrieve the heat flux in the crystallizer and the heat transfer coefficient in the secondary cooling zone. The complementary information, required for solving the investigated inverse problem, was delivered by the measurements of temperature at fixed points on the ingot. The elaborated method was based on two procedures: the implicit scheme of finite difference method supplemented by the procedure of correcting the field of temperature in the vicinity of liquidus and solidus curves, used for solving the appropriate direct problem (see Mochnacki et al. [28,29,30]) and the artificial bee colony (ABC) algorithm, applied to minimize the appropriate function expressing the error of the approximate solution (see Karaboga et al. [36,37,38]). The computational example illustrating the correctness and stability of the proposed approach is also presented in this study.

2. Formulation of the Problem

The objective of the investigation was the continuous casting of steel executed in a vertical device running in an undisturbed cycle. One assumes that the cooling conditions vary with reference to the direction of the forming ingot. One also assumes that the cooling conditions are the same along the entire perimeter of the created ingot. One presumes that the heat flows only in a direction perpendicular to the axis of the ingot. The last assumption follows from the fact that the amount of heat conducted in the direction of movement of the ingot is insignificant in comparison to the amount of heat conducted in the direction perpendicular to the ingot axis (see Mochnacki et al. [28]). During the process, the apparently steady temperature field, generated during the course of the undisturbed working cycle of the continuous casting device, was considered.

Under the above assumptions and because of the heat symmetry, the investigations can be reduced to the region of one quarter of the ingot as shown in Figure 1. Thus, the region of the ingot can be considered as the three-dimensional region $\mathsf{\Omega }=[0,a]\times [0,b]\times [0,z^{\ast }]\subset {\mathbb{R}}^3$. On the boundary of region $\mathsf{\Omega }$ the following subsets are separated

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_0& =\left\{(x,y,0);x\in [0,a],y\in [0,b]\right\},\hfill \\ \hfill {\mathsf{\Gamma }}_1& =\left\{(0,y,z);y\in [0,b],z\in (0,z^{\ast }]\right\},\hfill \\ \hfill {\mathsf{\Gamma }}_2& =\left\{(x,0,z);x\in [0,a],z\in (0,z^{\ast }]\right\},\hfill \\ \hfill {\mathsf{\Gamma }}_3& =\left\{(x,b,z);x\in [0,a],z\in (0,z_1)\right\},\hfill \\ \hfill {\mathsf{\Gamma }}_4& =\left\{(a,y,z);y\in [0,b],z\in (0,z_1)\right\},\hfill \\ \hfill {\mathsf{\Gamma }}_5& =\left\{(x,b,z);x\in [0,a],z\in [z_1,z^{\ast }]\right\},\hfill \\ \hfill {\mathsf{\Gamma }}_6& =\left\{(a,y,z);y\in [0,b],z\in [z_1,z^{\ast }]\right\},\hfill \end{array}$$

(1)

where the boundary conditions are defined.

In the scheme presented in Figure 1, point $z_1$ on the Z axis indicates the point at which the continuous casting device is divided into two parts: the crystallizer (for $z\in [0,z_1)$), where the casted material begins to solidify and from where it was moved next to the secondary cooling zone (for $z\in [z_1,z^{\ast }]$). The problem then consists of the determination of the cooling conditions for the continuous ingot, which were: the heat flux in the crystallizer (that is on boundaries ${\mathsf{\Gamma }}_3$ and ${\mathsf{\Gamma }}_4$) and the heat transfer coefficient in the secondary cooling zone (that is on boundaries ${\mathsf{\Gamma }}_5$ and ${\mathsf{\Gamma }}_6$). The cooling conditions should be identified so that the temperature in the selected points of the ingot would take the given values $T(x_i,y_i,z_j)=U_{ij},$ for $i=1,\dots ,N_1$ and $j=1,\dots ,N_2$, where $N_1$ means the number of sensors and $N_2$ denotes the number of measurements read from each sensor.

The solidifying material occupies region $\mathsf{\Omega }$ was divided into three subregions: the first was taken by the solid phase (for $T<T_S$), the second was taken by the liquid phase (for $T>T_L$) and finally the mushy zone (for $T\in [T_S,T_L]$) where both phases were mixed. Temperatures $T_L$ and $T_S$ denote the liquidus and solidus temperatures respectively, determining the beginning and the end of the solidification process. Under the assumption of neglecting the natural convection in the liquid phase, as well as the strain energy of the mushy zone, the following heat conduction equation, describing the apparently steady field of temperature, is defined (see Mochnacki et al. [28,29]):

$$wC\varrho \frac{\partial T(x,y,z)}{\partial z}=\frac{\partial }{\partial x}\left({\lambda }_{eff}\frac{\partial T(x,y,z)}{\partial x}\right)+\frac{\partial }{\partial y}\left({\lambda }_{eff}\frac{\partial T(x,y,z)}{\partial y}\right),$$

(2)

where T denotes the temperature, w describes the pulling rate (the cast slab shifts in the z direction), $\varrho$ is the mass density and ${\lambda }_{eff}$ is the effective thermal conductivity. The above equation C defines the substitute thermal capacity (or equivalent specific heat):

$$C=c-L\frac{\partial f_s}{\partial T},$$

(3)

where c and L are the specific heat and latent heat of fusion, respectively, and $f_s$ describes the volumetric solid state fraction. Equation (2) describes the field of temperature in the entire homogeneous region, that is in the liquid phase, mushy zone and solid phase.

The value of substitute thermal capacity depends on the form of function $f_s$ describing the volumetric solid state fraction. In turn, function $f_s$ varies in dependence on temperature, that is, for the liquidus and solidus temperatures one has:

$$f_s\left(T_L\right)=0\mathrm{and}{f}_s\left(T_S\right)=1,$$

respectively, whereas for $T\in (T_S,T_L)$ one can apply Scheil’s equation, according to the work by Santos et al. [22]:

$$f_s\left(T\right)=1-{\left(\frac{T_f-T}{T_f-T_L}\right)}^{\frac{1}{k_0-1}},$$

(4)

for modeling the macrosegregation in the steel, where $T_f$ denotes the melting temperature of the pure iron and $k_0$ is the partition coefficient describing the quotient of concentrations of the alloy in the solid phase and in the liquid phase.

After introducing the volumetric solid state fraction, the following form of the substitute thermal capacity can be taken

$$\begin{array}{c}\hfill C=\left\{\begin{array}{ccc}{c}_l\hfill & \hfill & T\ge T_L,\hfill \\ {\displaystyle c_{mz}-L\frac{\partial f_s}{\partial T}}\hfill & \hfill & T\in (T_S,T_L),\hfill \\ c_s\hfill & \hfill & T\le T_S,\hfill \end{array}\right.\end{array}$$

(5)

where $c_l$, $c_{mz}$ and $c_s$ denote the specific heat of the liquid phase, mushy zone and solid phase, respectively. One assumes that the value of specific heat in the mushy zone is expressed as dependent on the solid phase contribution $f_s$ as given below:

$$c_{mz}=c_l(1-f_s)+c_s{f}_s.$$

Similarly, the mass density, showing up in Equation (2), depends on temperature in the following way:

$$\varrho =\left\{\begin{array}{ccc}{\varrho }_l\hfill & \hfill & T\ge T_L,\hfill \\ {\varrho }_{mz}\hfill & \hfill & T\in (T_S,T_L),\hfill \\ {\varrho }_s\hfill & \hfill & T\le T_S,\hfill \end{array}\right.$$

(6)

where ${\varrho }_l$, ${\varrho }_{mz}$ and ${\varrho }_s$ mean the density of liquid phase, mushy zone and solid phase, respectively, under the assumption that:

$${\varrho }_{mz}={\varrho }_l(1-f_s)+{\varrho }_s{f}_s.$$

Finally, the effective thermal conductivity, appearing in Equation (2), is expressed by means of the relation (see Lally et al. [33], Choudhary et al. [34] and Oksman et al. [35]):

$${\lambda }_{eff}=\left\{\begin{array}{cc}{k}_c\lambda \left(T_S\right)\hfill & T\ge T_L,\hfill \\ \lambda \left(T_S\right)\left[1+(k_c-1){\left(\frac{T-T_S}{T_L-T_S}\right)}^2\right]\hfill & T\in (T_S,T_L),\hfill \\ \lambda \left(T\right)\hfill & T\le T_S,\hfill \end{array}\right.$$

(7)

where $k_c=3$ and

$$\lambda \left(T\right)=12.3167+0.0107534T+7.7098\times {10}^{-7}{T}^2$$

for $T\le T_S$ (see Miettinen [39]).

To complete Equation (2), the appropriate initial and boundary conditions are introduced on the respective parts of the boundary defined by Equation (1). Thus, on boundary ${\mathsf{\Gamma }}_0$ the following condition of the first kind must be satisfied

$$T(x,y,0)=T_0,$$

(8)

where $T_0>T_L$ denotes the pouring temperature.

On boundaries ${\mathsf{\Gamma }}_1$ and ${\mathsf{\Gamma }}_2$ the homogeneous condition of the second kind is given

$$-{\lambda }_{eff}\frac{\partial T(x,y,z)}{\partial n}=0.$$

(9)

Boundary conditions of the second kind are also defined on boundaries ${\mathsf{\Gamma }}_3$ and ${\mathsf{\Gamma }}_4$ representing the crystallizer:

$$\begin{array}{c}\hfill -{\lambda }_{eff}\frac{\partial T(x,y,z)}{\partial n}=q_3\left(z\right),\end{array}$$

(10)

$$\begin{array}{c}\hfill -{\lambda }_{eff}\frac{\partial T(x,y,z)}{\partial n}=q_4\left(z\right),\end{array}$$

(11)

where functions $q_3$ and $q_4$ describe the heat flux.

Finally, in the secondary cooling zone, that is on boundaries ${\mathsf{\Gamma }}_5$ and ${\mathsf{\Gamma }}_6$, the boundary conditions of the third kind are assumed

$$\begin{array}{c}\hfill -{\lambda }_{eff}\frac{\partial T(x,y,z)}{\partial n}={\alpha }_5\left(z\right)\left(T(x,y,z)-T^{\infty }\right),\end{array}$$

(12)

$$\begin{array}{c}\hfill -{\lambda }_{eff}\frac{\partial T(x,y,z)}{\partial n}={\alpha }_6\left(z\right)\left(T(x,y,z)-T^{\infty }\right),\end{array}$$

(13)

where functions ${\alpha }_5$ and ${\alpha }_6$ describe the equivalent heat transfer coefficient and $T^{\infty }$ denotes the temperature of environment (water or air). It is assumed that in the model investigated the value of the heat transfer coefficient in the respective zone includes all the factors influencing the cooling conditions in this zone (that is the roller temperature, cooling rates, etc.).

Values of functions $q_3$, $q_4$, ${\alpha }_5$ and ${\alpha }_6$ are unknown and the goal of the considered inverse problem is to identify them on the basis of the measurements of temperature read from the fixed points of the ingot.

3. The Solution Procedure

The unknown elements are retrieved on the basis of the temperature measurements in selected points of the domain:

$$T(x_i,y_i,z_j)=U_{ij},i=1,\dots ,N_1,j=1,\dots ,N_2,$$

(14)

where $N_1$ denotes the number of sensors and $N_2$ means the number of measurements taken from each sensor.

For the fixed form of functions $q_3$, $q_4$, ${\alpha }_5$ and ${\alpha }_6$ the problem, seen in Equations (2)–(13), is the direct solidification problem. Solution of the direct problem gives the values of temperature $T_{ij}=T(x_i,y_i,z_j)$ corresponding to the assumed form of reconstructed functions. By using the calculated values of temperature $T_{ij}$ and the given values $U_{ij}$ one can write the equation expressing the error of approximate solution:

$$J(q_3,q_4,{\alpha }_5,{\alpha }_6)=\sum _{i=1}^{N_1}\sum _{j=1}^{N_2}{\left(T_{ij}-U_{ij}\right)}^2.$$

(15)

Minimization of Equation (15) leads to determination of the unknown elements $q_3$, $q_4$, ${\alpha }_5$ and ${\alpha }_6$ so that the calculated values of temperature are as close as possible to the measured values.

The direct solidification problem, considered for the fixed form of functions $q_3$, $q_4$, ${\alpha }_5$ and ${\alpha }_6$, is solved by using the implicit scheme of the finite difference method supplemented by the procedure of correcting the field of temperature in the vicinity of liquidus and solidus curves (see Mochnacki et al. [28,29]). In the cast region, for the cross-section z, the mesh is introduced like the one presented in Figure 2. However, the size of the step along the Z axis is different for $z\in [0,z_1)$ (in the crystallizer) and for $z\in [z_1,z^{\ast }]$ (in the secondary cooling zone).

Value of temperature $T_{i,j}^k$ in node $(x_i,y_j,z_k)$ is determined by solving the proper system of equations received in result of applying the finite difference method and the proper approximation of the boundary conditions. The matrix of coefficients obtained for this system of algebraic equations is the band matrix, therefore the successive over relaxation method was applied for solving this system in order to ensure the faster convergence of an iterative process (see for example Hadjidimos [40]). Next, the value of temperature $T_{i,j}^{k+1}$, calculated for node $(x_i,y_j,z_{k+1})$, is corrected in case of the phase change in node $(x_i,y_j,z_{k+1})$, according to the procedure described below.

First, one can discuss the situation when in node $(x_i,y_j,z_{k+1})$; the phase does not change. For example, one can assume that node $(x_i,y_j,z_k)$ is in the liquid phase, which means that $T_{i,j}^k>T_L$. Therefore, to execute the next step, the values of parameters for the liquid phase should be used. If the condition $T_{i,j}^{k+1}>T_L$ is still satisfied for the newly calculated temperature, it means that the node is still in the liquid phase, so the temperature $T_{i,j}^{k+1}>T_L$ is correctly determined.

However, if the inequality $T_{i,j}^{k+1}⩽T_L$ holds true, it means that the node $(x_i,y_j,z_{k+1})$ is in the mushy zone, so the value of newly calculated temperature requires correction. Change of the phase happens along the distance $\Delta z=z_{k+1}-z_k$. For part of this distance the values of parameters should be taken properly in the liquid phase, and for the other part, properly in the mushy zone. The process of cooling from temperature $T_{i,j}^k$ to temperature $T_{i,j}^{k+1}$ is connected with the change of enthalpy in control element $V_{i,j}$, which is executed in two stages: cooling from temperature $T_{i,j}^k$ to liquidus temperature $T_L$ and cooling from temperature $T_L$ to the sought value of temperature ${\overline{T}}_{i,j}^{k+1}$. Thus, the balance of enthalpy can be expressed:

$$\Delta H_i=C_l{\varrho }_l(T_{i,j}^k-T_L)\Delta V_{i,j}+C_{mz}{\varrho }_{mz}(T_L-{\overline{T}}_{i,j}^{k+1})\Delta V_{i,j},$$

where $C_l,$$C_{mz}$ and ${\varrho }_l,$${\varrho }_{mz}$ denote the substitute thermal capacity and density of the liquid phase and mushy zone respectively.

In general, the change of enthalpy in element $V_{i,j}$, associated with the process of cooling from temperature $T_{i,j}^k$ to temperature $T_{i,j}^{k+1}$, can be written in the form:

$$\Delta H_i=C_l{\varrho }_l(T_{i,j}^k-T_{i,j}^{k+1})\Delta V_{i,j}.$$

Comparing the two relations above, the following formula for the corrected value of temperature in node $(x_i,y_j,z_{k+1})$ is obtained:

$${\overline{T}}_{i,j}^{k+1}=T_L-\frac{C_l{\varrho }_l}{C_{mz}{\varrho }_{mz}}(T_L-T_{ij}^{k+1}).$$

The formula for the corrected value of the temperature in node $(x_i,y_j,z_{k+1})$, in the case when the node passes from the mushy zone to the solid phase, can be derived in a similar way and has the form:

$${\overline{T}}_{ij}^{k+1}=T_S-\frac{C_{mz}{\varrho }_{mz}}{C_s{\varrho }_s}(T_S-T_{ij}^{k+1}),$$

where $C_s$ and ${\varrho }_s$ denote the substitute thermal capacity and density of the solid phase respectively. The situation of passing from the liquid phase immediately to the solid phase is also hypothetically possible, but one can avoid this case by the correct selection of the mesh step along the Z axis.

4. The Optimization Procedure

For the optimizing Equation (15), the ABC algorithm is used. It is an algorithm heuristic in nature belonging to the group of artificial intelligence algorithms invented by Karaboga in 2005 [36] (see also Karaboga et al. [37,38]). The idea of the ABC algorithm is inspired by the behavior of bees exploring the environment around the hive in search of sources of nectar. When a bee discovers the nectar, it informs the other members of the swarm with the aid of a special kind of dance called the waggle dance. The waggle dance takes place in a special spot in the hive, where the dancing bee moves in a straight line at first and then returns to the starting point along a semicircle, once to the right side, next to the left side. The direction of the bee’s dance indicates the angle between the sun and the source of nectar. The time taken to move in the straight line refers to the distance between the hive and the source of nectar (each 75 milliseconds of straight movement is equal to 100 m in distance). The magnitude of the bee’s body vibration during the dance indicates the quality of the nectar. This particular behavior of bees was studied and described in the 1940s by Karl von Frish [41].

In the optimization algorithm, imitating the above described behavior, the population of artificial bees is divided into two equal parts of number $SN$: the bee-scouts exploring the environment in search of nectar and the bee-viewers waiting in the hive to watch the waggle dance. In the first part of the algorithm the bee-scouts locate the assumed number of sources of nectar ${\mathbf{x}}_i$ and make some number of control movements in order to check whether it is in the neighborhood of the selected point so that a better location can be found. In the second part of the algorithm, the bee-viewers choose the sources of nectar from among the points of the domain selected and modified by the bee-scouts using the appropriate probabilities. If the probability of choosing is greater then the quality of the source is better, that equates to the lowest value of the minimized equation in point ${\mathbf{x}}_i$ under examination. Next, the bee-viewers explore the selected sources by making a number of control movements in order to improve the quality of the localized source. Each cycle of the algorithm finishes by choosing the best source of nectar in the current cycle. A detailed description of the ABC algorithm is as follows.

Initial data:

• $SN$—number of the explored sources of nectar (number of bee-scouts, number of bee-viewers);
• D—dimension of the source ${\mathbf{x}}_i$, $i=1,\dots ,SN$;
• $lim$—number of “corrections” of the source location ${\mathbf{x}}_i$, we take $lim=SN\times D$;
• $MCN$—maximal number of cycles.

Initial step:

• Random selection of the initial locations of sources represented by the D-dimensional vectors ${\mathbf{x}}_i$, $i=1,\dots ,SN$.
• Calculation of values $J\left({\mathbf{x}}_i\right)$, $i=1,\dots ,SN$, of minimized equation.

Main part of the algorithm:

• Modification of the source locations by the bee-scouts:
  (a) Every bee-scout modifies location ${\mathbf{x}}_i$ according to formula:

  $$v_i^j=x_i^j+{\varphi }_{ij}(x_i^j-x_k^j),j\in \{1,\dots ,D\},$$

  where: $\left.\begin{array}{c}k\in \{1,\dots ,SN\},k\ne i,\hfill \\ {\varphi }_{ij}\in [-1,1].\hfill \end{array}\right\}$—randomly selected numbers.
  (b) If $J\left({\mathbf{v}}_i\right)\le J\left({\mathbf{x}}_i\right)$, then location ${\mathbf{v}}_i$ replaces ${\mathbf{x}}_i$. Otherwise, location ${\mathbf{x}}_i$ remains unchanged.
  Steps (a) and (b) are repeated $lim$ times, where $lim=SN\times D$.
• Calculation of probabilities $P_i$ for the locations ${\mathbf{x}}_i$ selected in step 1. We use the equation:

  $$P_i=\frac{fi{t}_i}{{\displaystyle \sum _{j=1}^{SN}}fi{t}_j},i=1,\dots ,SN,$$

  where: $fi{t}_i=\left\{\begin{array}{c}\frac{1}{1+J\left({\mathbf{x}}_i\right)}\mathrm{if}J\left({\mathbf{x}}_i\right)\ge 0,\hfill \\ 1+|J\left({\mathbf{x}}_i\right)|\mathrm{if}J\left({\mathbf{x}}_i\right)<0.\hfill \end{array}\right.$
• Every bee-viewer chooses one of the sources ${\mathbf{x}}_i$, $i=1,\dots ,SN$, with probability $P_i$. Of course, only one source can be chosen by a group of bees.
• Every bee-viewer explores the chosen source and modifies its location according to the procedure described in step 1.
• Selection of the ${\mathbf{x}}_{best}$ for the current cycle—the best source among the sources determined by the bee-viewers. If the current ${\mathbf{x}}_{best}$ is better than the one from the previous cycle, it is accepted as the ${\mathbf{x}}_{best}$ for the entire algorithm.
• If in step 1 the bee-scout did not improve location ${\mathbf{x}}_i$ (${\mathbf{x}}_i$ did not change), it leaves the source ${\mathbf{x}}_i$ and moves to the new one by using equation:

  $$x_i^j=x_{min}^j+{\varphi }_{ij}(x_{max}^j-x_{min}^j),j=1,\dots ,D,$$

  where ${\varphi }_{ij}\in [0,1],$ $x_{min}^j$ and $x_{max}^j$ denote the minimal and maximum value of the $j-$th coordinate in the current population respectively.

Steps 1–6 are repeated $MCN$ times.

5. Numerical Calculations

In this section the numerical example executed for verifying the elaborated procedure is described. The problem of continuous casting of steel is solved for the following values of parameters (Miettinen [39], Spinelli et al. [42] and Janik et al. [43]):

• dimensions of the cast $a=0.08$ [m], $b=0.08$ [m], $z^{\ast }=15.5$ [m],
• length of the crystallizer $z_1=0.5$ [m],
• $T_S=1672$K, $T_L=1728$K, $T_0=1823$K, $T_f=1809$K,
• $c_s=700$ [J/(kg K)], $c_l=800$ [J/(kg K)], ${\varrho }_s=7256$ [kg/m³], ${\varrho }_l=7011$ [kg/m³],
• $L=189,000$ [J/kg], $k_0=0.84$, $w=2.5$ [m/min].

The goal of the inverse problem being studied is to reconstruct the cooling conditions for the continuous ingot, that is the heat flux in the crystallizer (described by functions $q_3$ and $q_4$ in boundary conditions, Equations (10) and (11)) and the heat transfer coefficient in the secondary cooling zone (described by functions ${\alpha }_5$ and ${\alpha }_6$ in boundary conditions, Equations (12) and (13)), as well as the distribution of temperature inside the investigated region, based on the known measurements of temperature in selected points of the cast. The measurements of temperature were read from two or three thermocouples ($N_1=3$ or $N_1=2$) located in the neighborhood of point $(a,b)$ of the region being studied (see Figure 3). In the case of two sensors, the thermocouples numbered “1” and “3” were used. The distance between the successive measurements along the Z axis was equal to 0.01 m (which makes $N_2=1530$) or 0.02 m (which makes $N_2=765$).

In the experiment performed the exact values of the required parameters were known and they were equal to:

• $q_3=1.85\times {10}^6$ [W], $q_4=2.0\times {10}^6$ [W],
• ${\alpha }_5\left(z\right)=\left\{\begin{array}{cc}1000,\hfill & z\in (0.5,1.2],\hfill \\ 700,\hfill & z\in (1.2,3],\hfill \\ 300,\hfill & z\in (3,15.5]\hfill \end{array}\right.$
  ${\alpha }_6\left(z\right)=\left\{\begin{array}{cc}900,\hfill & z\in (0.5,1.2],\hfill \\ 600,\hfill & z\in (1.2,3],\hfill \\ 280,\hfill & z\in (3,15.5],\hfill \end{array}\right.$ (in unit [W/(m²K)]).

For constructing Equation (15) the exact values of temperature, calculated for the known values of the required parameters (presented above) and the values given random errors of 1%, 2% and 5% were used.

Solution of the inverse problem requires the multiple solutions of the direct problem for the procedure of minimizing the objective function, see Equation (15). To solve the direct problem the implicit scheme of the finite difference method completed with the procedure of correcting the temperature field in the neighborhood of the liquidus and solidus curves is used, as it is described in Section 3. In the region studied, 100 nodes were used along the X axis and 100 nodes along the Y axis, equally spaced (see Figure 2). The implicit difference scheme was applied along the Z axis for the mesh with step $\Delta z_c=0.001$ in the region of the crystallizer and $\Delta z_{scz}=0.01$ in the region of the secondary cooling zone. To avoid the inverse crime (see Kaipio et al. [44]) the input data (that is the simulated measurements) were generated for a much denser grid.

The boundary conditions are reconstructed in three ways. In the first approach (one step optimization) the heat fluxes in the crystallizer and the heat transfer coefficients in the secondary cooling zone were retrieved simultaneously, that is eight values were identified at the same time. In the second approach (two steps optimization) the optimization problem was divided into two steps. First the heat fluxes in the crystallizer was reconstructed, that is two parameters were identified. Next, the heat transfer coefficients in the secondary cooling zone were retrieved, that is, six parameters were identified, but in boundary conditions (Equations (10) and (11)) the values of heat flux, already calculated in the first step of the optimization procedure, were used. In the last variant (four steps optimization) the optimization was split into four steps. In this approach the values of the heat flux in the crystallizer and the values of the heat transfer coefficients in the secondary cooling zone were reconstructed consecutively. This meant that four optimization tasks were solved, in each one, two parameters were identified. The parameters calculated in the previously solved optimization tasks were used as the boundary conditions in the next optimization task.

For minimizing Equation (15) the ABC algorithm was applied. The procedure was executed for $SN=8$ bees and $MCN=25$ cycles in case of one step optimization and for $SN=7$ bees and $MCN=10$ cycles in the other two cases of optimization, which were the values giving the best results. On the basis of some previous test calculations and the experience taken from some previous works (see for example Hetmaniok et al. [45,46]) we decided that the bigger number of bees or cycles would extend the calculation time unnecessarily. Additionally, with regard to the heuristic nature of the ABC algorithm, which means that each execution of the procedure can give slightly different results, the calculations were evaluated 10 times and the best set of results were accepted as the reconstructed elements. The multiple launches of the procedure aimed to verify the correctness of the algorithm used, and this was proved by obtaining similar results in each execution of the procedure, obtaining values which were very close to the required optimal values.

5.1. One Step Optimization

First of all, the analysis of results received in the simultaneous reconstruction of all desired parameters will be presented.

Table 1. Reconstructed values of the unknown coefficients together with the absolute and relative errors of these reconstructions, the standard deviation $\sigma$ and the standard deviation ${\sigma }_p$ taken as the percentage of the best reconstructed value, obtained for the measurements read at every 0.01 m (one step optimization).

Noise	par.	Reconstructed	Absolute	Relative	Standard	Standard
		Value	Error	Error (%)	dev. $\mathit{\sigma }$	dev. ${\mathit{\sigma }}_{\mathit{p}}(\%)$
1%	$q_3$	1.834 $\times {10}^6$	$1.556\times {10}^4$	0.8484	9946.72	0.5422
	$q_4$	$2.022\times {10}^6$	$2.207\times {10}^4$	$1.0914$	$6322.86$	$0.3127$
	${\alpha }_{(5,1)}$	$910.43$	$10.4334$	$1.1460$	$18.4110$	$2.0222$
	${\alpha }_{(5,2)}$	$602.75$	$2.7504$	$0.4563$	$3.1201$	$0.5176$
	${\alpha }_{(5,3)}$	$280.37$	$0.3717$	$0.1326$	$1.6610$	$0.5924$
	${\alpha }_{(6,1)}$	$1010.03$	$10.0299$	$0.9930$	$14.7367$	$1.4590$
	${\alpha }_{(6,2)}$	$700.70$	$0.6957$	$0.0993$	$2.3147$	$0.3303$
	${\alpha }_{(6,3)}$	$300.17$	$0.1740$	$0.0580$	$1.5986$	$0.5326$
2%	$q_3$	$1.846\times {10}^6$	$3619.52$	$0.1960$	$1.263\times {10}^4$	$0.6840$
	$q_4$	$2.008\times {10}^6$	$8055.89$	$0.4012$	$1.078\times {10}^4$	$0.5369$
	${\alpha }_{(5,1)}$	$910.97$	$10.9667$	$1.2039$	$11.7559$	$1.2905$
	${\alpha }_{(5,2)}$	$597.12$	$2.8787$	$0.4821$	$1.7792$	$0.2980$
	${\alpha }_{(5,3)}$	$280.62$	$0.6193$	$0.2207$	$0.1615$	$0.0575$
	${\alpha }_{(6,1)}$	$1006.92$	$6.9151$	$0.6868$	$9.5205$	$0.9455$
	${\alpha }_{(6,2)}$	$704.94$	$4.9449$	$0.7015$	$1.2405$	$0.1760$
	${\alpha }_{(6,3)}$	$300.11$	$0.1120$	$0.0373$	$0.0786$	$0.0262$
5%	$q_3$	$1.849\times {10}^6$	$1288.39$	$0.0697$	$1.006\times {10}^4$	$0.5443$
	$q_4$	$2.030\times {10}^6$	$3.027\times {10}^4$	$1.4911$	$5852.90$	$0.2883$
	${\alpha }_{(5,1)}$	$932.75$	$32.7515$	$3.5113$	$12.1607$	$1.3037$
	${\alpha }_{(5,2)}$	$588.66$	$11.3405$	$1.9265$	$1.8191$	$0.3090$
	${\alpha }_{(5,3)}$	$277.70$	$2.2993$	$0.8280$	$0.3707$	$0.1335$
	${\alpha }_{(6,1)}$	$973.88$	$26.1209$	$2.6822$	$8.7825$	$0.9018$
	${\alpha }_{(6,2)}$	$713.35$	$13.3498$	$1.8714$	$1.6295$	$0.2284$
	${\alpha }_{(6,3)}$	$303.03$	$3.0318$	$1.0005$	$0.5110$	$0.1686$

shows the reconstructed values of the required parameters, that is the values of heat flux in the crystallizer ($q_3$ and $q_4$) and the values of the heat transfer coefficient in the secondary cooling zone (three values of functions ${\alpha }_5$ and ${\alpha }_6$), obtained for all three considered perturbations of measurement data read at every 0.01 m, together with the absolute and relative errors of these reconstructions. In this table, the values of standard deviations obtained in ten executions of the procedure were also collected. When analyzing the results, one may observe that in all investigated cases of input data the obtained results are satisfactory and comparable with the known values of retrieved parameters. In most cases, the relative errors of the reconstructions are lower than the input data errors. However, the calculation error is just slightly higher than 1% for three parameter values, reconstructed by using the measurements with 1% noise. The reconstructed values are the best values of results obtained over ten executions of the procedure, created by the set of values for which the smallest value of Equation (15) is reached. However, all the obtained results are very similar which testifies to the stability of the applied method. A reflection of this is the obtained values of standard deviations, in particular referenced to the best determined value of the analyzed parameter.

In case of the exact input data the maximum reconstruction error of the required parameter usually does not exceed the value 0.3%. Only the parameters ${\alpha }_{(5,1)}$ and ${\alpha }_{(6,1)}$ are found with slightly bigger errors. The value of the reconstruction error is mainly due to the mesh density used in the calculations. Increasing the mesh density causes the error to decrease for the required parameters. However it increases the calculation time. The number of iterations in the successive over relaxation method, needed for solving one direct problem varied between 25 and 95, with an average value of 46.

Figure 4 presents the comparison of the reconstructed and exact distribution of temperature calculated at control point “2” (that is in the point of thermocouple “2” located in the corner of the region) for the measurements read at every 0.01 m and with 5% error. The absolute error of this reconstruction is also presented in this figure. Similar results are received for other control points (that is the locations of thermocouples “1” and “3”, see Figure 5) and the other input data perturbations. This conclusion is illustrated by

Table 2. Errors of temperature reconstruction in the control points obtained for measurements read at every 0.01 m (one step optimization).

Noise	${\mathbf{\Delta }}_{\mathit{m}\mathit{a}\mathit{x}}\left(\mathit{K}\right)$	${\mathbf{\Delta }}_{\mathit{a}\mathit{v}\mathit{g}}\left(\mathit{K}\right)$	${\mathit{\delta }}_{\mathit{m}\mathit{a}\mathit{x}}(\%)$	${\mathit{\delta }}_{\mathit{a}\mathit{v}\mathit{g}}(\%)$
$1\%$	$3.3233$	$0.1732$	$0.2576$	$0.0178$
$2\%$	$1.9579$	$0.2615$	$0.1697$	$0.0277$
$5\%$	$4.6963$	$0.5999$	$0.4624$	$0.0640$

, in which the maximum and mean errors of reconstructing the temperature in control points are collected. In case of the exact input data the maximum errors of temperature reconstruction do not exceed the value 1.72 K (0.19%), whereas the mean error is equal to 0.11 K (0.01%).

Finally, in Figure 6 one may observe the development of the reconstructed process of solidification. In the successive figures one can see the selected stages of solidification of the steel ingot, processing along the Z axis, with the marked boundaries of the liquid and solid phases and the mushy zone in between. In cross-section $z=10$ m only two phases: mushy zone and solid phase are present. Next, in cross-section $z=15$ m, all the material had already solidified.

In when the number of control points was two times smaller, the obtained results are equally good. The reconstructed values of desired parameters, together with the reconstruction errors and the standard deviations of the determined values, are presented in

Table 3. Reconstructed values of the unknown coefficients together with the absolute and relative errors of these reconstructions, the standard deviation $\sigma$ and the standard deviation ${\sigma }_p$ taken as the percent of the best reconstructed value, obtained for the measurements read at every 0.02 m (one step optimization).

Noise	par.	Reconstructed	Absolute	Relative	Standard	Standard
		Value	Error	Error $(\%)$	dev. $\mathit{\sigma }$	dev. ${\mathit{\sigma }}_{\mathit{p}}(\%)$
1%	$q_3$	$1.842\times {10}^6$	$7672.31$	$0.4164$	$882.58$	$0.0479$
	$q_4$	$2.002\times {10}^6$	$1858.39$	$0.0928$	$741.35$	$0.0370$
	${\alpha }_{(5,1)}$	$925.98$	$25.9794$	$2.8056$	$1.0897$	$0.1177$
	${\alpha }_{(5,2)}$	$601.34$	$1.3383$	$0.2226$	$0.4613$	$0.0767$
	${\alpha }_{(5,3)}$	$279.47$	$0.5251$	$0.1879$	$0.0929$	$0.0332$
	${\alpha }_{(6,1)}$	$999.20$	$0.7975$	$0.0798$	$1.0663$	$0.1067$
	${\alpha }_{(6,2)}$	$702.91$	$2.9077$	$0.4137$	$0.4330$	$0.0616$
	${\alpha }_{(6,3)}$	$300.83$	$0.8332$	$0.2770$	$0.0877$	$0.0292$
2%	$q_3$	$1.864\times {10}^6$	$1.435\times {10}^4$	$0.7699$	$4732.91$	$0.2539$
	$q_4$	$2.007\times {10}^6$	$6876.33$	$0.3426$	$4309.06$	$0.2147$
	${\alpha }_{(5,1)}$	$892.95$	$7.0514$	$0.7897$	$3.9057$	$0.4374$
	${\alpha }_{(5,2)}$	$610.34$	$10.3407$	$1.6943$	$0.7972$	$0.1306$
	${\alpha }_{(5,3)}$	$281.56$	$1.5585$	$0.5535$	$0.1192$	$0.0423$
	${\alpha }_{(6,1)}$	$1021.22$	$21.2161$	$2.0775$	$3.7143$	$0.3637$
	${\alpha }_{(6,2)}$	$693.92$	$6.0760$	$0.8756$	$0.7999$	$0.1153$
	${\alpha }_{(6,3)}$	$299.10$	$0.8977$	$0.3001$	$0.1143$	$0.0382$
5%	$q_3$	$1.877\times {10}^6$	$2.732\times {10}^4$	$1.4550$	$4150.99$	$0.2211$
	$q_4$	$2.026\times {10}^6$	$2.643\times {10}^4$	$1.3041$	$5883.80$	$0.2904$
	${\alpha }_{(5,1)}$	$908.71$	$8.7088$	$0.9584$	$6.5646$	$0.7224$
	${\alpha }_{(5,2)}$	$596.23$	$3.7687$	$0.6321$	$2.5080$	$0.4206$
	${\alpha }_{(5,3)}$	$288.11$	$8.1087$	$2.8145$	$0.7397$	$0.2568$
	${\alpha }_{(6,1)}$	$972.48$	$27.5202$	$2.8299$	$7.8579$	$0.8080$
	${\alpha }_{(6,2)}$	$718.63$	$18.6317$	$2.5927$	$2.2665$	$0.3154$
	${\alpha }_{(6,3)}$	$290.08$	$9.9194$	$3.4195$	$0.7261$	$0.2503$

. In most cases the reconstruction errors are lower or just slightly higher than the input data errors. Only in one case, that is for 1% input data perturbation, the reconstruction error is about 2.8%. This situation is probably caused by the change of boundary conditions, because in the assumed model the step changes in cooling conditions when the material leaves the crystallizer. The bigger mesh density results in a decrease in the reconstruction error of this parameter. The values of temperature are in each case retrieved well, with small errors not exceeding 0.18% (see

Table 4. Errors of temperature reconstruction in the control points obtained for measurements read at every 0.02 m (one step optimization).

Noise	${\mathbf{\Delta }}_{\mathit{m}\mathit{a}\mathit{x}}\left(\mathit{K}\right)$	${\mathbf{\Delta }}_{\mathit{a}\mathit{v}\mathit{g}}\left(\mathit{K}\right)$	${\mathit{\delta }}_{\mathit{m}\mathit{a}\mathit{x}}(\%)$	${\mathit{\delta }}_{\mathit{a}\mathit{v}\mathit{g}}(\%)$
$1\%$	$3.7039$	$0.2172$	$0.3698$	$0.0218$
$2\%$	$3.3010$	$0.4484$	$0.3277$	$0.0493$
$5\%$	$9.1192$	$1.5355$	$0.8978$	$0.1737$

and Figure 7).

In the next experiment, the number of sensors decreased from three to two ($N_1=2$). The measurements taken from thermocouples “1” and “3” were used in calculations. The parameters describing the boundary conditions, which were reconstructed in this case, were a little bit worse than in the experiments with three sensors, but still the maximum reconstruction errors were kept at the level of input data errors and the mean errors were significantly lower. In Figure 8, the relative errors of temperature reconstruction in the control points obtained for measurement data taken from two and three thermocouples are given. Data read from lower numbered sensors give a slightly worse reconstruction of the temperature. For input data with a 5% error the maximum relative reconstruction error does not exceed 0.91% and the mean error does not exceed 0.11%, which corresponds to the absolute reconstruction errors equal to 11.95 and 0.99 K, respectively. For the lower input data perturbations, the reconstruction errors are also significantly lower than the input data errors.

5.2. Two Steps Optimization

In this section, the results obtained in the case when the optimization task was divided into two steps are discussed. In the first step, the heat fluxes in the crystallizer were retrieved, which means that in this optimization task two parameters ($q_3$ and $q_4$) were reconstructed. In the second step the heat transfer coefficients in the secondary cooling zone were retrieved, which means that in this optimization task six parameters (${\alpha }_{(5,i)}$ and ${\alpha }_{(6,i)}$ for $i=1,2,3$) were reconstructed. In the second optimization step the heat fluxes calculated in the first optimization step were used in the boundary conditions (Equations (10) and (11)).

Table 5. Reconstructed values of the unknown coefficients together with the absolute and relative errors of these reconstructions, the standard deviation $\sigma$ and the standard deviation ${\sigma }_p$ taken as the percentage of the best reconstructed value, obtained for the measurements read at every 0.01 m (two steps optimization).

Noise	par.	Reconstructed	Absolute	Relative	Standard	Standard
		Value	Error	Error $(\%)$	dev. $\mathit{\sigma }$	dev. ${\mathit{\sigma }}_{\mathit{p}}(\%)$
1%	$q_3$	$1.823\times {10}^6$	$2.749\times {10}^4$	$1.5085$	$3172.28$	$0.1741$
	$q_4$	$2.017\times {10}^6$	$1.661\times {10}^4$	$0.8236$	$2336.80$	$0.1159$
	${\alpha }_{(5,1)}$	$917.27$	$17.2744$	$1.8832$	$5.8286$	$0.6354$
	${\alpha }_{(5,2)}$	$602.31$	$2.3108$	$0.3836$	$4.0404$	$0.6708$
	${\alpha }_{(5,3)}$	$280.42$	$0.4244$	$0.1513$	$1.7928$	$0.6393$
	${\alpha }_{(6,1)}$	$1014.80$	$14.8006$	$1.4585$	$5.7709$	$0.5687$
	${\alpha }_{(6,2)}$	$700.72$	$0.7211$	$0.1029$	$3.5224$	$0.5027$
	${\alpha }_{(6,3)}$	$299.74$	$0.2606$	$0.0870$	$1.8072$	$0.6029$
2%	$q_3$	$1.829\times {10}^6$	$2.091\times {10}^4$	$1.1432$	$1.330\times {10}^4$	$0.7271$
	$q_4$	$2.034\times {10}^6$	$3.393\times {10}^4$	$1.6680$	$1.570\times {10}^4$	$0.7717$
	${\alpha }_{(5,1)}$	$906.18$	$6.1777$	$0.6817$	$10.2675$	$1.1331$
	${\alpha }_{(5,2)}$	$597.75$	$2.2537$	$0.3770$	$1.1967$	$0.2002$
	${\alpha }_{(5,3)}$	$278.78$	$1.2154$	$0.4359$	$0.9103$	$0.3265$
	${\alpha }_{(6,1)}$	$1006.02$	$6.0167$	$0.5981$	$6.1809$	$0.6144$
	${\alpha }_{(6,2)}$	$706.03$	$6.0266$	$0.8536$	$1.6220$	$0.2297$
	${\alpha }_{(6,3)}$	$299.35$	$0.6547$	$0.2187$	$0.9129$	$0.3050$
5%	$q_3$	$1.947\times {10}^6$	$9.719\times {10}^4$	$4.9911$	$1.046\times {10}^4$	$0.5371$
	$q_4$	$1.967\times {10}^6$	$3.282\times {10}^4$	$1.6686$	$9631.15$	$0.4896$
	${\alpha }_{(5,1)}$	$902.44$	$2.4358$	$0.2699$	$9.8713$	$1.0939$
	${\alpha }_{(5,2)}$	$588.57$	$11.4293$	$1.9419$	$4.1490$	$0.7049$
	${\alpha }_{(5,3)}$	$276.07$	$3.9273$	$1.4226$	$2.0609$	$0.7465$
	${\alpha }_{(6,1)}$	$980.44$	$19.5591$	$1.9949$	$9.8610$	$1.0058$
	${\alpha }_{(6,2)}$	$716.11$	$16.1138$	$2.2502$	$6.1077$	$0.8529$
	${\alpha }_{(6,3)}$	$302.38$	$2.3774$	$0.7862$	$2.2218$	$0.7348$

shows the reconstructed values of the required parameters, that is, the values of heat flux in the crystallizer and the values of the heat transfer coefficient in the secondary cooling zone, obtained for all three studied perturbations of measured data read at every 0.01 m, together with the absolute and relative errors of these reconstructions and the values of standard deviations obtained in ten executions of the procedure. In this approach, the boundary conditions were identified, in general, with the reconstruction errors not exceeding the input data errors. For only 1% of the input data perturbations, the reconstruction error is slightly higher, that is, in three cases, but still it does not exceed the value of 1.89%. In all launches of the procedure, very similar values of the reconstructed parameters were obtained, which is confirmed by the standard deviations collected in Table 5.

In Figure 9 the relative errors of temperature reconstruction at the measurement points obtained for the input data with a 5% error are presented. The maximum absolute reconstruction error is not higher than 14.6 K, and the mean error is not higher than 1.3 K. The maximum and mean absolute and relative reconstruction errors for all considered input data perturbations are collected in

Table 6. Errors of temperature reconstruction in the control points obtained for measurements read at every 0.01 m (two steps optimization).

Noise	${\mathbf{\Delta }}_{\mathbf{max}}\left(\mathit{K}\right)$	${\mathbf{\Delta }}_{\mathbf{avg}}\left(\mathit{K}\right)$	${\mathit{\delta }}_{\mathbf{max}}(\%)$	${\mathit{\delta }}_{\mathbf{avg}}(\%)$
$1\%$	$5.6717$	$0.2935$	$0.4396$	$0.0307$
$2\%$	$4.0571$	$1.2762$	$0.3145$	$0.1490$
$5\%$	$14.5948$	$1.2628$	$1.1148$	$0.1394$

.

5.3. Four Steps Optimization

In the last investigated variant, the optimization task was divided into four steps. First, the values of the heat flux in the crystallizer were reconstructed and then the values of heat transfer coefficient in the successive parts of the secondary cooling zone were reconstructed consecutively, one by one. This means that four optimization tasks were solved and each optimization task consists in the reconstruction of two parameters. The parameters calculated in the previous tasks were then used as the boundary conditions for solving the next tasks.

The reconstructed values of required parameters, together with the reconstruction errors and the values of standard deviations obtained in ten executions of the procedure, are collected in

Table 7. Reconstructed values of the unknown coefficients together with the absolute and relative errors of these reconstructions, the standard deviation $\sigma$ and the standard deviation ${\sigma }_p$ taken as the percentage of the best reconstructed value, obtained for the measurements read at every 0.01 m (four steps optimization).

Noise	par.	Reconstructed	Absolute	Relative	Standard	Standard
		Value	Error	Error $(\%)$	dev. $\mathit{\sigma }$	dev. ${\mathit{\sigma }}_{\mathit{p}}(\%)$
1%	$q_3$	$1.829\times {10}^6$	$2.131\times {10}^4$	$1.1651$	$5539.93$	$0.3029$
	$q_4$	$2.022\times {10}^6$	$2.172\times {10}^4$	$1.0741$	$3358.56$	$0.1661$
	${\alpha }_{(5,1)}$	$909.82$	$9.8206$	$1.0794$	$1.9821$	$0.2179$
	${\alpha }_{(5,2)}$	$607.97$	$7.9739$	$1.3115$	$1.2275$	$0.2019$
	${\alpha }_{(5,3)}$	$279.65$	$0.3493$	$0.1249$	$0.2090$	$0.0747$
	${\alpha }_{(6,1)}$	$1014.33$	$14.3269$	$1.4124$	$1.7708$	$0.1746$
	${\alpha }_{(6,2)}$	$704.32$	$4.3231$	$0.6138$	$1.1336$	$0.1609$
	${\alpha }_{(6,3)}$	$299.49$	$0.5092$	$0.1700$	$0.2107$	$0.0703$
2%	$q_3$	$1.835\times {10}^6$	$1.488\times {10}^4$	$0.8109$	$7087.61$	$0.3862$
	$q_4$	$2.011\times {10}^6$	$1.146\times {10}^4$	$0.5698$	$4808.33$	$0.2390$
	${\alpha }_{(5,1)}$	$904.63$	$4.6349$	$0.5123$	$5.8069$	$0.6419$
	${\alpha }_{(5,2)}$	$605.08$	$5.0801$	$0.8396$	$0.2380$	$0.0393$
	${\alpha }_{(5,3)}$	$276.94$	$3.0649$	$1.1067$	$0.0845$	$0.0305$
	${\alpha }_{(6,1)}$	$1000.28$	$0.2803$	$0.0280$	$1.2442$	$0.1244$
	${\alpha }_{(6,2)}$	$709.23$	$9.2259$	$1.3008$	$0.1926$	$0.0272$
	${\alpha }_{(6,3)}$	$300.14$	$0.1446$	$0.0482$	$0.0343$	$0.0114$
5%	$q_3$	$1.923\times {10}^6$	$7.265\times {10}^4$	$3.7787$	$4.002\times {10}^4$	$2.0816$
	$q_4$	$1.986\times {10}^6$	$1.374\times {10}^4$	$0.6919$	$3.452\times {10}^4$	$1.7382$
	${\alpha }_{(5,1)}$	$907.08$	$7.0801$	$0.7805$	$17.7788$	$1.9600$
	${\alpha }_{(5,2)}$	$592.28$	$7.7205$	$1.3035$	$0.6301$	$0.1064$
	${\alpha }_{(5,3)}$	$276.24$	$3.7631$	$1.3623$	$0.1692$	$0.0613$
	${\alpha }_{(6,1)}$	$955.16$	$44.8357$	$4.6940$	$12.7364$	$1.3334$
	${\alpha }_{(6,2)}$	$722.38$	$22.3761$	$3.0976$	$0.1149$	$0.0159$
	${\alpha }_{(6,3)}$	$303.36$	$3.3588$	$1.1072$	$0.1176$	$0.0388$

. One may also see that in this approach the boundary conditions are very well retrieved and the successive executions of the procedures give very similar results, which is confirmed by the values of standard deviation. Figure 10 shows the relative errors of temperature reconstruction in the control points obtained for input data perturbed by 5% error. The maximum absolute reconstruction error is no higher than 11.3 K, and the mean error is no higher than 1 K. The maximum and mean absolute and relative reconstruction errors for all levels of the input data perturbations are presented in

Table 8. Errors of temperature reconstruction in the control points obtained for measurements read at every 0.01 m (four steps optimization).

Noise	${\mathbf{\Delta }}_{\mathit{x}\mathit{a}\mathit{x}}\left(\mathit{K}\right)$	${\mathbf{\Delta }}_{\mathit{a}\mathit{v}\mathit{g}}\left(\mathit{K}\right)$	${\mathit{\delta }}_{\mathit{x}\mathit{a}\mathit{x}}(\%)$	${\mathit{\delta }}_{\mathit{a}\mathit{v}\mathit{g}}(\%)$
$1\%$	$4.3580$	$0.6313$	$0.3386$	$0.0703$
$2\%$	$3.8065$	$0.4706$	$0.3819$	$0.1856$
$5\%$	$11.2115$	$0.9849$	$0.9085$	$0.1039$

.

In Figure 11 the errors of temperature reconstruction obtained in three investigated variants of optimization are compared. Slightly lower reconstruction errors are received in the case where optimization is achieved in one step. However, the division of optimization tasks into a few steps influences the time of calculations significantly. In the case of one step optimization the calculations took about 4875 min, in the case of two steps optimization about 1640 min, whereas in case of four steps optimization, only about 730 min. For performing the one step optimization we used the ABC algorithm with 8 bees ($SN=8$) and 25 cycles ($MCN=25$), whereas when performing the two other variants of optimization we needed only 7 bees ($SN=7$) and 10 cycles ($MCN=10$).

6. Conclusions

This paper presents the procedure for solving the inverse problem of the continuous casting of steel consisting of reconstruction, on the basis of known measurements of temperature, the cooling conditions of the process, that is the heat flux in the crystallizer and the heat transfer coefficient in the secondary cooling zone. The proposed approach was based on the mathematical model of solidification in the temperature interval which includes the process of macrosegregation consisting of the variation of the concentration of the alloy component in the forming cast. Scheil’s equation was used to define the concentration. This is usually applied to modeling macrosegregation in stainless steel. In the solution procedure the finite difference method supplemented by the appropriate procedure of correcting the temperature field was used for solving the direct problem associated with the inverse problem and the ABC algorithm for minimizing the function expressing the error of the approximate solution. The proposed procedure was investigated with regard to its stability and the precision of the results obtained.

The calculations presented in this study show a very good approximation of the exact solution by the reconstructed values obtained for various input data errors. In each investigated case of input data perturbations the errors of the identified coefficients were smaller, or at least comparable, with the errors of input data and the errors of temperature reconstruction were insignificant. Moreover, the results obtained in multiple execution of the procedure were very similar. Summing up, the proposed procedure constitutes an effective tool for solving the inverse problem of this problem.

Comparing the three variants of optimization we conclude that the best match to the input data was ensured by the first variant, when all parameters were determined simultaneously. The disadvantage of such an approach is, however, the time needed to execute the calculations, which was significantly bigger than in case of the other optimization variants. The fastest calculations were performed in the third optimization variant, when the optimization task was divided into four steps and in each step only two parameters were reconstructed. The results obtained by using this approach were only slightly worse than the best results. Therefore, if a short calculation time is important and a slightly less precise reconstruction of the boundary condition is acceptable, then we advise that the optimization task should be divided into a few “smaller” optimization components.

The novelty of this paper lies in the elaboration of an algorithm serving as a solution to the inverse problem for the model of continuous casting with the inclusion of the macrosegregation phenomenon and the effective thermal conductivity for molten steel as well as the examination of accuracy and stability of the developed algorithm. Another novelty is in the comparison of the three optimization variants differing in the number of optimization steps, which enables a reduction in the calculation time.

The developed algorithm gives the possibility to select the cooling conditions of the ingot in such a way that the entire solidification process runs in an expected, predetermined way. Thanks to this, the steel producers can protect the produced ingots from the formation of various kinds of defects. Depending on the chosen optimization variant one can moderate the speed of calculations at the expense of a slight worsening in the precision of the result.

The paper presents the results for a particular type of steel. The change of the steel composition causes a change in the values of the parameters in the model (liquidus and solidus temperatures, density, specific heat etc.). Such changes of the parameter values does not influence the work of the developed algorithm. The procedure remains equally precise and stable regardless of the type of steel.